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CLASSICAL RELATIONS AND FUZZY RELATIONS
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報告流程 卡氏積 (Cartesian Product) 明確關係 (Crisp Relations)
Cardinality Operations Properties 合成 (Composition)
模糊關係 (Fuzzy Relations) Cardinality Operations Properties Fuzzy Cartesian Product and Compositon Noninteractive Fuzzy Sets
Crisp Tolerance and Equivalence Relations Fuzzy Tolerance and Equivalence Relations Value Assignments
Cosine Amplitude Max-min Method Other Similarity Methods
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Cartesian Product
Producing ordered relationships among sets
X × Y = {(x,y)│x X, y Y}∈ ∈ All the Ar = A
A1 × A2 × ……. × Ar = Ar
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Cartesian Product
Example 3.1 Set A = { 0,1 } Set B = { a, b, c }
A × B = {(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}
B × A = {(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)}
A × A = A2 = {(0,0),(0,1),(1,0),(1,1)}
B × B = B2 ={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}
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Crisp Relations
Measure by characteristic function : χ X × Y = {(x,y)│x X, y Y}∈ ∈
Binary relation χX×Y(x,y)= 1, (x,y) X × Y ∈ 0, (x,y) X × Y χR(x,y)= 1, (x,y) X × Y ∈ 0, (x,y) X × Y
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Crisp Relations
a b c
1
R = 2
3
111
111
111
Sagittal diagram Relation Matrix
EX: X={1,2,3} Y={a,b,c}
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Crisp Relations
Example 3.2 ( 一 )
X={1,2} Y={a,b} 1 a Locations of zero 2 b
R={(1,a),(2,b)} R X × Y ( 二 )
A={0,1,2} UA : universal relation IA : identity relation 以 A2 為例
UA = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)} IA = {(0,0),(1,1),(2,2)}
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Crisp Relations
Example 3.3 Continous universes R={(x,y) | y ≥ 2x, x X, y Y}∈ ∈ χR(x,y)= 1, y ≥ 2x
0, Y < 2x
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Cardinality of Crisp Relations
X : n elements Y : m elements
n X : the cardinality of X
n Y : the cardinality of Y Cardinality of the relation
n X × Y = nX * nY
power set The cardinality : P(X × Y) n P(X × Y) = 2(nXnY)
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Operations on Crisp Relations
Union
Intersection
Complement
Containment
Identity(Ø → O and X → E)
)],(),,(max[),(:),( yxyxyxyxSR SRSRSR
)],(),,(min[),(:),( yxyxyxyxSR SRSRSR
),(1),(:),( yxyxyxR RRR
),(),(:),( yxyxyxSR SRR
()
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Properties of Crisp Relations 交換律 (Commutative law)
結合律 (Associative law)
分配律 (Distributive law)
乘方 (Involution) 冪等律 (Idempotence)
狄摩根定律 (De Morgan’s law)
排中律 (Low of Excluded Middle)UAA
ABBAABBA ,
CBACBACBA )()(
)()()( CABACBA
AAAAAA ,
,, BABABABA
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Composition
R={(X1,Y1),(X1,Y3),(X2,Y4)}
S={(Y1,Z2),(Y3,Z2)}
Composition oeration Max-min composition
T=R 。 S
Max-product comositon T=R 。 S
)),(),((),( zyyxzx SRYy
T
)),(),((),( zyyxzx SRYy
T
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Composition Example 3.4
Max-min composition y1 y2 y3 y4 z1 z2
R= x1 S= y1
x2 y2
x3 y3 y4 z1 z2
T= x1
x2 x3
0000
0001
1010
00
01
00
01
00
00
01
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Fuzzy Relations
Membership function Interval [0,1] Cartesian space X × Y =>
Cardinality of Fuzzy Relations Universe is infinity
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Operations on Fuzzy Relations
Union
Intersection
Complement
Containment
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Properties of Fuzy Relations
排中律 (Low of Excluded Middle) 在 Fuzzy 集合中並不成立 !
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Fuzzy Cartesian Product
Cartesian product space
Fuzzy relation has membership function
Example 3.5
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Fuzzy Composition
Fuzzy max-min composition
Fuzzy max-product composition
不論 crisp 或 fuzzy 的 composition
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Fuzzy Composition Example 3.6
X={x1,x2} Y={y1,y2} Z={z1,z2,z3}
Max-min composition
Max-product compositon
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Noninteractive Fuzzy Sets Fuzzy set on the Cartesian space X =X1 × X2
noninteractive
interactive
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Noninteractive Fuzzy Sets
Example 3.7
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Noninteractive Fuzzy Sets Example 3.7( 續 )
Cartesian product
120
2.0
100
0.1
60
7.0
30
3.0)(% seseR
120
1.0
100
0.1
80
8.0
60
6.0
40
4.0
20
2.0)(% aaI
1800
15.0
1500
0.1
1000
67.0
500
33.0)( rpmN
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Noninteractive Fuzzy Sets Example 3.7(續 )
Max-min composition
Example 3.8
Max-min composition
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Noninteractive Fuzzy Sets Example 3.9
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Tolerance and Equivalence relations
自返性 (reflexivity)
對稱性 (symmetry)
傳遞性 (transitivity)
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Crisp Eqivalence Relation
自返性 (reflexivity) (xi,xi) R or
對稱性 (symmetry) (xi,xj) R (xj,xi) R
or 傳遞性 (transitivity)
(xi,xj) R and (xj,xk) R (xi,xk) R
or
1),( iiR xx
),(),( ijRjiR xxxx
1),(1),(),( kiRkjRjiR xxxxandxx
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Crisp Tolerance Relation
Also called proximity relation Only the reflexivity and symmetry Can be reformed into an equivalence
relation By at most (n-1) compositions with itself
RRRRR n 111
11 ....
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Crisp Tolerance Relation
Example 3.10 X={x1,x2,x3,x4,x5}={Omaha, Chicago, Rome,
London, Detroit}
R1 does not properties of transitivity e.g. (x1,x2) R1 (x2,x5) R1 but (x1,x5) R1
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Crisp Tolerance Relation
Example 3.10(續 ) R1 can become an equivalence relation through two
compositions
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Fuzzy tolerance and equivalence relations
自返性 (reflexivity)
對稱性 (symmetry)
傳遞性 (transitivity)
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Fuzzy tolerance and equivalence relations
Equivalence relations Fuzzy tolerance relation Can be
reformed into an equivalence relation By at most (n-1) compositions with itself
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Fuzzy tolerance and equivalence relations
Example 3.11
It is not transitive
One composition
Reflexive and symmetric
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Fuzzy tolerance and equivalence relations
Example 3.11(續 )
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Value assignments Cartesian product Closed-from expression
Simple observation of a physical process No variation
model the process crisp relation Y= f(X)
Lookup table Variability exist
Membership values on the interval [0,1] Develop a fuzzy relation
Linguistic rules of knowledge If-then rules
Classification Similarity methods in data manipulation
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Cosine Amplitude
X={x1,x2,….,xn}
xi={ }
miii xxx ,....,, 21
))((
||
1 1
22
1
m
k
m
k jkik
m
k jkikij
xx
xxr
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Cosine Amplitude
Example 3.12
r12=0.836
Regions x1 x2 x3 x4 x5
Xi1—Ratio with no damage 0.3 0.2 0.1 0.7 0.4
Xi2—Ratio with medium damage 0.6 0.4 0.6 0.2 0.6
Xi3—Ratio with serious damage 0.1 0.4 0.3 0.1 0
))((
||3
1
3
1
22
3
1
k k jkik
k jkikij
xx
xxr
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Cosine Amplitude Example 3.12(續 )
Tolerance relation
Equivalence relation
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Max-min Method
rij= where i, j =1,2,…n
Example 3.13 Reconsider Example 3.12
Tolerance relation
m
kjkik
m
kjkik
xx
xx
1
1
),max(
),min(
3
1
3
112
))4.0,1.0max(),4.0,6.0max(),2.0,3.0(max(
))4.0,1.0min(),4.0,6.0min(),2.0,3.0(min(
k
kr
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Summary
Q & A